where /' refers to the derivative of / with respect to its argument x. Finally, for future reference note that

For the study of the structure of hadrons, we are interested in positive energy solutions (antiquarks will be described by their positive energy charge conjugate states) for which m is both less than and greater than the bound state energy E. There will be two kinds of solutions, depending on the parity (or the sign of k) of the state. If A; > 0, i = k, and for solutions in the vicinity of the origin we must choose f(r) = fk(x) = Njk(kor) , (6.31a)

where N is a normalization constant and x = k0r. The other solution, proportional to nfc, is singular, and hence unacceptable. For a normalizable solution, we must also choose the one which approaches zero as r —> oo, or f(r) = fk(x) = Nhi1](iK0r) , (6.31b)

*A good reference for special functions is Abramowitz and Stegun (1964).

6.3 HYDROGEN-LIKE ATOMS The indicial equation for v is obtained when n — 0:

(v + k)A0 - ZaB0 = 0 ZaA0 + (v- k)B0 = 0 . To have a non-trivial solution (i.e., Ao or B0 / 0) requires k2 + (Za)2 = 0

Now the reduced wave function goes like p" at the origin and is singular if v is negative. In order for it to be normalizable, this singularity cannot be stronger than which implies the condition u > — Hence, negative values of i> must be rejected because, even for the smallest value of \k\ (|fc| = 1), negative i> are less than — A (unless Z is very large, and we will not discuss such extreme cases).

Hence _

and eliminating An-i and from the coupled Eqs. (6.50) gives the following relation for Bn in terms of An:

The recursion relations for An can now be found by substituting (6.54) into the coupled Eqs. (6.50), in+l

(u + n + 1 - k + Zae) (2u + 2n + Za (e - {)) (v + n - k + Zae) ((i/ + n 4-1)2 - k2 + (Za)2)

The eigenvalue condition emerges from the recursion relation (6.55). First, note that as n —> oo, the ratio An+\/An —» 2/(n + 1). For comparison,

where the ratio of successive terms of this comparison series is

By the ratio test, the series for Jr and qr will therefore go like fR —» e2p e~p = ep ,

This singularity is very weak, and the solution is still integrable near the origin. The lower component is very much smaller (by a factor of \Za) than the upper component. Hence the relativistic solution differs from the non-relativistic solution only to order Za, or at very short distances.

This concludes our study of first quantized relativistic equations. In the next chapter, we begin the study of field theories based on these equations.


6.1 Consider quarks confined in a spherical volume, as discussed in Sec. 6.2. Suppose that it requires energy to "make" a volume in which the quarks can move freely, so that the total energy of n non-interacting quarks inside a volume R is where, for ground state quarks, x0 = 2.04, and B is the energy density of the empty volume.

(a) Minimize the energy with respect to R and show that min ~ V 4ttB

Suppose the proton (mass 940 MeV) is made of three quarks. What is its radius? What is the mass and radius of a qq system? How does this compare with masses of the known mesons (it = 140 MeV, p = 770 MeV, w ~ 783 MeV)?

(b) Suppose the confined quark has a rest mass m, ^ 0. Find an equation for its energy if it is confined in a spherical volume of radius R.

6.2 Find the solution for the first excited state of the MIT bag.

6.3 Suppose a massless quark moves under the influence of a SHO potential with both scalar and vector terms,

V(r) = Ajr2 + 0 (V0 + A2r2) , where the term proportional to Ai is the fourth component of a vector, the second term (with the Dirac matrix (3) is a scalar mass term, and Vo, Ai, and A2 are constants.

(a) Find the correct coupled equations for the upper and lower radial functions fk{r) and gk(r) of the Dirac wave function of such a state.


(b) Find the single second order equation for fk{r).

(c) Choose the constants V0, A1; and A2 so that this equation reduces to a Schrodinger equation for a particle moving in a pure simple harmonic potential. Find the ground state energy and the Dirac wave function for the ground state of a massless quark moving in such a potential. Discuss the significance of your result.

6.4 A massless spin | particle moves in a one-dimensional scalar potential of the form

= VQ z < -R and R < z , where the constant V0 is large and positive.

(a) Write down the correct Dirac equation for the motion of this particle.

(b) Show that the equation found in part (a) is invariant under the parity transformation z —>• —z.

(c) Solve the equation for the ground state energy and wave function of the trapped particle. Take the limit Vo —> oo, and sketch the solution for this case. Comment on any interesting features which the solution possesses.

6.5 A Dirac particle of mass m and positive charge e scatters from the one-dimensional high barrier shown in Fig. 4.1.

(a) Write down the Dirac equation which correctly describes the scattering if the potential energy eV is the zeroth component of a four-vector (a Coulomblike interaction).

(b) Write down the Dirac equation which correctly describes the scattering if the potential energy is a scalar (invariant under all Lorentz transformations).

(c) Consider solutions in region II of the form where x and 77 are two-component spinors, E > 0 is fixed, and eV > E + m. Solve the Dirac equation in region II for the two cases described above, and discuss the nature of the solutions. Can particles propagate in region II?

(d) Find the full solution for both of the cases described in (a) and (b), and discuss the time evolution of a positive energy state which is localized at large negative 2 at large negative t and approaches the barrier. Show that the norm is conserved in both cases, and discuss your results.




Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS Copyright© 2004 WILEY-VCH Verlag GmbH


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